Real Gel′fand‐Mazur division algebras

Abstract

We show that the complexification of a real locally pseudoconvex (locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra (A, τ) is a complex locally pseudoconvex (resp., locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra and all elements in the complexification of a commutative real exponentially galbed algebra (A, τ) with bounded elements are bounded if the multiplication in (A, τ) is jointly continuous. We give conditions for a commutative strictly real topological division algebra to be a commutative real Gel′fand‐Mazur division algebra.